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Mirrors > Home > MPE Home > Th. List > uspgrushgr | Structured version Visualization version GIF version |
Description: A simple pseudograph is an undirected simple hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 15-Oct-2020.) |
Ref | Expression |
---|---|
uspgrushgr | ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2726 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 1, 2 | isuspgr 28920 | . . . 4 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
4 | ssrab2 4072 | . . . . 5 ⊢ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}) | |
5 | f1ss 6787 | . . . . 5 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅})) | |
6 | 4, 5 | mpan2 688 | . . . 4 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅})) |
7 | 3, 6 | biimtrdi 252 | . . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
8 | 1, 2 | isushgr 28829 | . . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
9 | 7, 8 | sylibrd 259 | . 2 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph)) |
10 | 9 | pm2.43i 52 | 1 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 {crab 3426 ∖ cdif 3940 ⊆ wss 3943 ∅c0 4317 𝒫 cpw 4597 {csn 4623 class class class wbr 5141 dom cdm 5669 –1-1→wf1 6534 ‘cfv 6537 ≤ cle 11253 2c2 12271 ♯chash 14295 Vtxcvtx 28764 iEdgciedg 28765 USHGraphcushgr 28825 USPGraphcuspgr 28916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-nul 5299 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fv 6545 df-ushgr 28827 df-uspgr 28918 |
This theorem is referenced by: uspgrupgrushgr 28945 usgredgedg 28995 vtxdusgrfvedg 29257 1loopgrvd2 29269 isomuspgr 47079 |
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